In [1]}, Huneke and Wiegand showed that two \(R\) –modules \(M\)  and \(N\)  over complete intersection rings satisfy the depth formula provided
\(\mbox{Tor}\,_i^R(M,N)=0\)  for \(i>0\) . In [2]}, Iyengar showed that the depth formula holds for two \(R\) –modules \(M\)  and \(N\) , provided one of the modules has finite complete intersection dimension and \(\mbox{Tor}\,_i^R(M,N)=0\)  for all \(i>0\) .
In [3]}, Araya and Yoshino generalized Auslander's original result. More precisely, they proved that the formula (REF ) holds provided one of the modules has finite complete intersection dimension and \(\mbox{Tor}\,^R_i(M,N)=0\)  for \(i\gg 0\) .
In [4]}, Bergh and Jorgensen proved that the depth formula holds in certain cases over Cohen-Macaulay rings, provided one of the modules has reducible complexity and \(\mbox{Tor}\,_i^R(M,N)=0\)  for \(i>0\) .
